arXiv subjects

Tomasz Piasecki

Publications and source records attributed to Tomasz Piasecki.

24 records · Page 2Linked to original sources

arXiv2016

Steady transport equation in Sobolev-Slobodetskii spaces

We show existence of a regular solution in Sobolev-Slobodetskii spaces to stationary transport equation with inflow boundary condition in a bounded domain $Ω\subset \mathbb{R}^2$. Our result is subject to quite general constraint on the shape of the boundary around the points where the characteristics become tangent to the boundary which applies in particular to piecewise analytical domains. Our result gives a new insight on the issue of boundary singularity for the inflow problem for stationary transport equation, solution of which is crucial for investigation of stationary compressible Navier-Stokes equations with inflow/outflow.

math.AP↗

arXiv2012

Compressible perturbation of Poiseuille type flow

The paper examines the issue of stability of Poiseuille type flows in regime of compressible Navier-Stokes equations in a three dimensional finite pipe-like domain. We prove the existence of stationary solutions with inhomogeneous Navier slip boundary conditions admitting nontrivial inflow condition in the vicinity of constructed generic flows. Our techniques are based on an application of a modification of the Lagrangian coordinates. Thanks to such approach we are able to overcome difficulties coming from hyperbolicity of the continuity equation, constructing a maximal regularity estimate for a linearized system and applying the Banach fixed point theorem.

math.AP↗

arXiv2012

Strong solutions to the Navier-Stokes-Fourier system with slip-inflow boundary conditions

We consider a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part. We show the existence of a strong solution provided the data are close to a constant, but nontrivial flow with sufficiently large dissipation in the energy equation.

math.AP↗

arXiv2009

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain $Ω= Ω_0 \times (0,L) \in \mathbb{R}^3$. We show existence of a solution $(v,ρ) \in W^2_p(Ω) \times W^1_p(Ω)$, where $v$ is the velocity of the fluid and $ρ$ is the density, that is a small perturbation of a constant flow $(\bar v \equiv [1,0,0], \bar ρ\equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar v,\bar ρ)$. The term $u \cdot \nabla w$ in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence $(v^n,ρ^n)$ that is bounded in $W^2_p(Ω) \times W^1_p(Ω)$ and satisfies the Cauchy condition in a larger space $L_{\infty}(0,L;L_2(Ω_0))$ what enables us to deduce that the weak limit of a subsequence of $(v^n,ρ^n)$ is in fact a strong solution to our problem.

math-ph↗

arXiv2009

Steady compressible Nevier-Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain in $\mathbf{R^2}$. We show existence of a strong solution $(v,ρ) \in W^2_p(Q) \times W^1_p(Q)$ that is a small perturbation of a constant flow $(\bar v \equiv [1,0],\bar ρ\equiv 1)$. We also show that this solution is unique in a class of small perturbations of the constant flow $(\bar v,\bar ρ)$. In order show the existence of the solution we adapt the techniques know from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.

math-ph↗

arXiv2008

Steady compressible Oseen flow with slip boundary conditions

We prove the existence of solution in a class H^2(Ω) x H^1(Ω) to steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with the boundary of class H^{5/2}. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to a problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under additional assumption on the geometry of the boundary.

math.AP↗